Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle

x2+y2+2⁢x+4⁢y=5⇒(x+1)2+(y+2)2=10,

from which it is frequently easier to read off important information (the center, radius, etc.)

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Completing the square can also be used to find the extremal value
of a quadratic polynomial [2] without calculus.
Let us illustrate this for the polynomialp⁢(x)=4⁢x2+8⁢x+9.
Completing the square yields

p⁢(x)

=

(2⁢x+2)2-4+9

=

(2⁢x+2)2+5

≥

5,

since (2⁢x+2)2≥0. Here, equality holds if and
only if x=-1.
Thus p⁢(x)≥5 for all x∈ℝ, and p⁢(x)=5 if and only if
x=-1.
It follows that p⁢(x) has a global minimum at x=-1, where p⁢(-1)=5.